Question

A box is to be made where the material for the sides and the lid cost​ $0.20 per square foot and the cost for the bottom is ​$0.65 per square foot. Find the dimensions of a box with volume 10 cubic feet that has minimum cost.

Answer 1

The dimension of the box is l×w×h = (2.274 × 2.274 × 1.934) ft³

Explanation:

From the given information;

Let a be the cost of the box

Let b be one side of the square base ; &

h to be the height of the box

We know that the volume of the box = 10 cubic feet

Then;

a²h = 10

h = 10/a²

The base = (0.65)a²

The top = (0.2)a²

The side = (0.2) a × 25/a²

= 5/a

For the four sides of the box now ;

= (0.2) 4a × 25/a²

= 0.8 × 25/a

= 20 /a

The total cost of the box is:

b = 0.65a² + 0.2a² + 20 /a

b = 0.85 a² + 20 /a

Taking differential of b with respect to a ;we have:

db/da = 1.7a - 1/a²(20) = 0

1.7 a³ - 20 = 0

1.7 a³ = 20

a³ = 20/1.7

a³ = 11.77

a =
`\sqrt[3]{11.77}`

a = 2.274 ft

Thus; the cost for the base of the box = (0.65)a²

the cost for the base of the box =(0.65) × ( 2.274)²

the cost for the base of the box = 3.362

The top of the box = (0.2)a²

The top of the box = (0.2)× ( 2.274)²

The top of the box = 1.034

The four sides of the box = 20 /a

The four sides of the box = 20/2.274

The four sides of the box = 8.795

the total cost = b = 0.85 a² + 20 /a

the total cost = 0.85 (2.274)² + 20 /2.274

the total cost = 4.395 + 8.795

the total cost = 13.19

Recall that:

the volume of the box = 10 cubic feet

Then;

a²h = 10

h = 10/a²

h = 10/ 2.274²

h 1.934

The dimension of the box is l×w×h = (2.274 × 2.274 × 1.934) ft³